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7
P
- Factorization of complete bipartite multigraphs
U. S. Rajput and Bal Govind Shukla*
Department of mathematics and astronomy, Lucknow University, Lucknow, India
E-mail: [email protected], [email protected]
(Received on: 25-11-11; Accepted on: 17-12-11)
________________________________________________________________________________________________
ABSTRACT
2p
P
-factorization of a complete bipartite graph for p an integer was studied by Wang [1]. Further, Beiliang [2]extended the work of Wang [1], and studied the
P
2k -factorization of complete bipartite multigraphs. For even value ofk in
P
k-factorization, the spectrum problem is completely solved [1, 2, 3]. However for odd value of k i.e.P
3,P
5,P
7and
P
9, the path factorization have been studied by a number of researchers [4, 5, 6, 7]. Again,P
3-factorizations ofcomplete bipartite multigraphs and symmetric complete bipartite multi-digraphs were studied by Wang and Beiliang [8]. In the present paper, we study
P
7- factorization of complete bipartite multigraphs and show that the necessary andsufficient conditions for the existence of
P
7- factorization of complete bipartite multigraph are:(1)
4
n
≥
3 ,
m
(2)
4
m
≥
3 ,
n
(3)
m
+ ≡
n
0(mod 7),
(4)
7
λ
mn
/[6(
m
+
n
)]
is an integer.Mathematics Subject Classification: 68R10, 05C70, 05C38.
Key words: Complete bipartite Graph, Factorization of Graph, Spanning Graph.
________________________________________________________________________________________________
1. INTRODUCTION:
Let
K
m n, be the complete bipartite graph with two partite set havingm
andn
vertices. The graphλ
K
m n, is disjointunion of
λ
graphs, each isomorphic toK
m n, . A subgraphF
ofλ
K
m n, is called a spanning sub graph ofλ
K
m n, ifF
contains all vertices ofλ
K
m n, . For positive integerK
, a path onK
-vertices is denoted byP
k. AP
k-factor of,
m n
K
λ
is a spanning subgraphF
ofλ
K
m n, such that every component ofF
is aP
k, and every pair ofP
k has novertex in common. A
P
k-factorization ofλ
K
m n, is a set of edge-disjointP
k-factors ofλ
K
m n, which is a partition ofthe set of edges of
λ
K
m n, . The multigraphλ
K
m n, is calledP
k-factorable whenever it has aP
k-factorization.In this paper we are discussing the necessary and sufficient conditions for the existence of a
P
7−
factorization ofcomplete bipartite multigraph
λ
K
m n, . LetP
7 be the path on seven vertices andλ
K
m n, isK
m n, in which every edgeis taken
λ
times. A spanning subgraphF
ofλ
K
m n, is called aP
7-factor if each component ofF
is isomorphic to7
P
. Ifλ
K
m n, is expressed as an edge disjoint sum ofP
7-factor, then this sum is called aP
7-factorization ofλ
K
m n, . ________________________________________________________________________________________________- . &/0, &/01
© 2011, IJMA. All Rights Reserved 2791
2. MAIN RESULT:
The necessary and sufficient conditions for the existence of a
P
7−
factorization of complete bipartite multigraph,
m n
K
λ
are given in theorem 2.1, below.Theorem: 2.1
λ
K
m n, has aP
7−
factorization if and only if (1)4
n
≥
3 ,
m
(2)
4
m
≥
3 ,
n
(3)
m
+ ≡
n
0(mod 7),
(4)
7
λ
mn
/[6(
m
+
n
)]
is an integer.Proof: Let
λ
K
m n, is factorized intor
number ofP
7−
factors, and t be the number of components of eachP
7−
factor. Then
(
)
7
m n
t
=
+
and7
.
[6(
)]
mn
r
m n
λ
=
+
Hence conditions (3) and (4) are necessary.
Among these
t
components, letx
andy
be the number of components whose end points are inY
andX
,respectively. Then we has
3
x
+
4
y
=
m
and4
x
+
3
y
=
n
.
Hence(4
3 )
7
m
n
x
=
−
and(4
3 )
.
7
n
m
y
=
−
From0
≤ ≤
x
m
and0
≤
y
≤
n
,
we have3
n
≤
4
m
and3
m
≤
4 .
n
Conditions (1) and (2) are, therefore necessary. Now we prove the following existence theorem, which is used later in this paperTheorem: 2.2 If
λ
K
m n, has aP
7−
factorization, thenλ
K
sm sn, has aP
7−
factorization for every positive integers
.Proof: Let
K
s s, is 1-factorable [9], and {H1, H2 … Hs} be a 1-factorization of it. For each i with 1 is
, replaceevery edge of Hi with a
λ
K
m n, to get a spanning sub graph Gi ofλ
K
sm sn, such that the Gi’s {1 is
} are pair wiseedge disjoint and there sum is
λ
K
sm sn, . Sinceλ
K
m n, isP
7−
factorable, therefore Gi is alsoP
7−
factorable, and hence,,
sm sn
K
λ
is alsoP
7−
factorable.Theorem: 2.3 If
λ
K
m n, has aP
7−
factorization, thens K
λ
m n, has aP
7−
factorization for every positive integers
.Proof: Construct a
P
7−
factorization ofλ
K
m n, repeatedlys
number of times. Then we have aP
7−
factorization of,
m n
s K
λ
.Now we will prove theorem 2.1. There are three cases to consider,
Case: 1 (
(4
m
=
3 ) :
n
In this case, from theorem 2.2 and theorem 2.3λ
K
3 ,4n n has aP
7−
factorization.Case: 2
(4
n
=
3 ) :
m
obviously,λ
K
3 ,4m m has aP
7−
factorization.Case: 3
(4
m
>
3
n
and4
n
>
3 ) :
m
In this case, let(4
3 )
,
7
n
m
a
=
−
(4
3 )
,
7
m
n
b
=
−
,
7
m n
t
=
+
and7
.
[6(
)]
mn
r
m n
λ
=
+
Then from conditions (1)-(4) in theorem 2.1,a b t
, ,
andr
are integers, and 0 <a
< m and 0 <b
< n. We have3
a
+
4
b
=
m
and4
a
+
3
b
=
n
. Hence2 (
)
.
6(
)
ab
r
a b
a b
λ
λ
=
+
+
+
Let6(
)
,
ab
z
a b
λ
=
+
which is a positive integer.And let gcd (3
a
, 4b) = d, 3a
= dp
, 4b = dq, where gcd (p, q) = 1. Thendq
is even and.
[6(
)]
dpq
z
p
q
λ
=
- . &/0, &/01
© 2011, IJMA. All Rights Reserved 2792 These equalities imply the following equalities.
6(4
3 )
,
p
q z
d
pq
λ
+
=
6(
)(4
3 )
,
p
q
p
q z
m
pq
λ
+
+
=
(16
9 )(4
3 )
,
2
p
q
p
q z
n
pq
λ
+
+
=
(
)(16
9 )
,
p
q
p
q z
r
pq
+
+
=
2 (4
3 )
,
p
p
q z
a
pq
λ
+
=
3 (4
3 )
.
2
q
p
q z
b
pq
λ
+
=
Here,
t
= the number of copies ofP
7in any factor,r
= the number ofP
7−
factor in the factorization,a
= the number of copies ofP
7 with its endpoints inY
in a particularP
7−
factor (type M),b
= the number of copies ofP
7 with its endpoints inX
in a particularP
7−
factor (type W),c
= the total number of copies ofP
7 in the whole factorization.The following lemma can be verified.
Lemma: 2.1 Let
a
,b
,
p
and q be positive integers, if gcd( , )
p q
= 1 then gcd(
p
+
q pq
,
)
=
1
and if gcd(
ap bq
,
)
=
1,
then gcd(
ap
+
bq pq
,
)
=
1.
By using
p
,q
,
andd
,
the parametersm
andn
, satisfying conditions (1)-(4) in theorem 2.1 can be expressed asfollows:
Lemma: 2.2
(1) If gcd
( , 9)
p
=
1 and gcd(q, 16) = 1 then there exist a positive integers
such that12(
)(4
3 )
,
m
=
p q
+
p
+
q s
λ
n
=
(16
p
+
9 )(4
q
p
+
3 )
q s
λ
,
4 (4
3 )
,
a
=
p
p
+
q s
λ
b
=
3 (4
q
p
+
3 )
q s
λ
,
2(
)(16
9 ) .
r
=
p
+
q
p
+
q s
(2) If gcd
( , 9)
p
=
1 and gcd(q, 16) = 2, Let q=2q1. Then there exist a positive integers
such that1 1
6(
2 )(2
3 )
,
m
=
p
+
q
p
+
q s
λ
n
=
(8
p
+
9 )(2
q
1p
+
3 )
q s
1λ
,
1
2 (2
3 )
,
a
=
p
p
+
q s
λ
b
=
3 (2
q
1p
+
3 )
q s
1λ
,
1 1
(
2 )(8
9 ) .
r
=
p
+
q
p
+
q s
(3) If gcd
( , 9)
p
=
1 and gcd(q, 16) = 4, Let q=4q2. Then there exist a positive integers
such that2 2
6(
4
)(
3 )
,
m
=
p
+
q
p
+
q s
λ
n
=
2(4
p
+
9
q
2)(
p
+
3 )
q s
2λ
,
2
2 (
3 )
,
a
=
p p
+
q s
λ
b
=
6 (
q p
2+
3 )
q s
2λ
,
2 2
(
4
)(4
9 ) .
- . &/0, &/01
© 2011, IJMA. All Rights Reserved 2793
(4) If gcd
( , 9)
p
=
1 and gcd(q, 16) = 8, let q=8 .q3 Then there exist a positive integers
such that3 3
3(
8 )(
6 )
,
m
=
p
+
q
p
+
q s
λ
n
=
2(2
p
+
9 )(
q
3p
+
6 )
q s
3λ
,
3
(
6 )
,
a
=
p p
+
q s
λ
b
=
6 (
q p
3+
3 )
q s
3λ
,
3 3
(
8 )(2
9 ) .
r
=
p
+
q
p
+
q s
(5) If gcd
( , 9)
p
=
1 and gcd(q, 16) = 8, let q=16q4. Then there exist a positive integers
such that4 4
3(
16 )(
12 )
,
m
=
p
+
q
p
+
q s
λ
n
=
4(
p
+
9 )(
q
4p
+
12 )
q s
4λ
,
4
(
12
)
,
a
=
p p
+
q s
λ
b
=
12
q p
4(
+
12
q s
4)
λ
,
4 4
2(
16 )(
9 ) .
r
=
p
+
q
p
+
q s
(6) If gcd
( , 9)
p
=
3 and gcd(q, 16) = 1, let p=3 .p1 Then there exist a positive integers
such that1 1
12(3
)(4
)
,
m
=
p
+
q
p
+
q s
λ
n
=
3(16
p
1+
3 )(4
q
p
1+
q s
)
λ
,
1 1
12
(4
)
,
a
=
p
p
+
q s
λ
b
=
3 (4
q
p
1+
q s
)
λ
,
1 1
2(3
)(16
3 ) .
r
=
p
+
q
p
+
q s
(7) If gcd
( , 9)
p
=
3 and gcd(q, 16) = 2, letq
=
2
q
1and p=3 .p1 Then there exist a positive integers
such that1 1 1 1
6(3
2 )(2
)
,
m
=
p
+
q
p
+
q s
λ
n
=
3(8
p
1+
3 )(2
q
1p
1+
q s
1)
λ
,
1 1 1
6
(2
)
,
a
=
p
p
+
q s
λ
b
=
3 (2
q
1p
1+
q s
1)
λ
,
1 1 1 1
(3
2 )(8
3 ) .
r
=
p
+
q
p
+
q s
(8) If gcd
( , 9)
p
=
3 and gcd(q, 16) = 4, letq
=
4
q
2 and p=3 .p1 Then there exist a positive integers
such that1 2 1 2
6(3
4 )(
)
,
m
=
p
+
q
p
+
q s
λ
n
=
6(4
p
1+
3 )(
q
2p
1+
q s
2)
λ
,
1 1 2
6
(
)
,
a
=
p p
+
q s
λ
b
=
6 (
q p
2 1+
q s
2)
λ
,
1 2 1 2
(3
4
)(4
3 ) .
r
=
p
+
q
p
+
q s
(9) If gcd
( , 9)
p
=
3 and gcd(q, 16) = 4, letq
=
8
q
3 and p=3 .p1 Then there exist a positive integers
such that1 3 1 3
3(3
8 )(
2 )
,
m
=
p
+
q
p
+
q s
λ
n
=
6(2
p
1+
3 )(
q
3p
1+
2 )
q s
3λ
,
1 1 3
3 (
2 )
,
a
=
p p
+
q s
λ
b
=
6 (
q p
3 1+
2 )
q s
3λ
,
1 3 1 3
(3
8 )(2
3 ) .
r
=
p
+
q
p
+
q s
(10) If gcd
( , 9)
p
=
3 and gcd (q, 16) = 16, letq
=
16
q
4and p=3 .p1 Then there exist a positive Integers
such that1 4 1 4
3(3
16 )(
4 )
,
m
=
p
+
q
p
+
q s
λ
n =12(p +3q )(p +4q )s/λ
1 4 1 4 ,1 1 4
3 (
4 )
,
a
=
p p
+
q s
λ
b
=
12
q p
4(
1+
4
q s
4)
λ
,
1 4 1 4
2(3
16 )(
3 ) .
r
=
p
+
q
p
+
q s
(11) If gcd
( , 9)
p
=
9 and gcd (q, 16) = 1. Let p = 9p2. Then there exist a positive integers
such that2 2
4(9
)(12
)
,
m
=
p
+
q
p
+
q s
λ
n
=
3(16
p
2+
q
)(12
p
2+
q s
)
λ
,
2 2
12
(12
)
,
a
=
p
p
+
q s
λ
b
=
q
(12
p
2+
q s
)
λ
,
2 2
2(9
)(16
) .
r
=
p
+
q
p
+
q s
(12) If gcd
( , 9)
p
=
9 and gcd (q, 16) = 2. Let p = 9p2 andq
=
2 .
q
1 Then there exist a positive integers
such that2 1 2 1
2(9
2 )(6
)
,
- . &/0, &/01
© 2011, IJMA. All Rights Reserved 2794
2 2 1
6
(6
)
,
a
=
p
p
+
q s
λ
b
=
q
1(6
p
2+
q s
1)
λ
,
2 1 2 1
(9
2 )(8
3 ) .
r
=
p
+
q
p
+
q s
(13) If gcd
( , 9)
p
=
9 and gcd (q, 16) = 4. Let p = 9p2 andq
=
4 .
q
2 Then there exist a positive integers
such that2 2 2 2
2(9
4
)(3
)
,
m
=
p
+
q
p
+
q s
λ
n
=
6(4
p
2+
q
2)(3
p
2+
q s
2)
λ
,
2 2 2
6
(3
)
,
a
=
p
p
+
q s
λ
b
=
2
q
2(3
p
2+
q s
2)
λ
,
2 2 2 2
(9
4
)(4
) .
r
=
p
+
q
p
+
q s
(14) If gcd
( , 9)
p
=
9 and gcd (q, 16) = 8. Let p = 9p2 andq
=
8 .
q
3 Then there exist a positive integers
such that2 3 2 3
(9
8 )(3
2 )
,
m
=
p
+
q
p
+
q s
λ
n
=
6(2
p
2+
q
3)(3
p
2+
2 )
q s
3λ
,
2 2 3
3
(3
2 )
,
a
=
p
p
+
q s
λ
b
=
2 (3
q
3p
2+
2 )
q s
3λ
,
2 3 2 3
(9
8 )(2
) .
r
=
p
+
q
p
+
q s
(15) If gcd
( , 9)
p
=
9 and gcd (q, 16) = 16. Let p = 9p2 andq
=
16 .
q
4 Then there exist a positive integers
such that2 4 2 4
(9
16
)(3
4
)
,
m
=
p
+
q
p
+
q s
λ
n
=
12(
p
2+
q
4)(3
p
2+
4
q s
4)
λ
,
2 2 4
3
(3
4 )
,
a
=
p
p
+
q s
λ
b
=
4
p
4(3
p
2+
4 )
q s
4λ
,
2 4 2 4
2(9
16
)(
) .
r
=
p
+
q
p
+
q s
Proof: We assume that gcd(p, q) = 1, gcd(p, 9) = 1 and gcd(q, 16) = 1 hold.
Then gcd(16p + 9q, 2) = gcd(4p + 3q, 2) = 1 and gcd(16p, 9q) = gcd(4p, 3q) = 1 hold. From lemma 2.2, we get
(16
9 )(4
3 )
2
p
q
p
q z
n
pq
λ
+
+
=
, which is an integer.Therefore
z
2
pq
must be an integer.Let
s
=
z
2
pq
, then the equalities in (1) hold. Similarly we can prove the other equalities of lemma 2.2.Below in lemma 2.3 and 2.4, we are giving the direct constructions of graphs for particular values of
m
andn
given in Case 1 and 8 of lemma 2.2. The value of s is taken as 1.Lemma: 2.3 For any positive integer p and q, let
m
=
6(
p
+
2 )(2
q
p
+
3 )
q
λ
andn
=
(8
p
+
9 )(2
q
p
+
3 )
q
λ
.
Thenλ
K
m n, has aP
7−
factorization.Proof: Let
a
=
2 (2
p
p
+
3 )
q
λ
,b
=
3 (2
q
p
+
3 )
q
λ
,r
=
(
p
+
2 )(8
q
p
+
9 )
q
andr
1=
(
p
+
2 )
q
,2
(8
9 )
r
=
p
+
q
. Let X and Y be two partite set ofλ
K
m n, and set, 1 0
, 2 0
{
:1
,1
},
{
:1
,1
},
i j
i j
X
x
i
r
j
m
Y
y
i
r
j
n
=
≤ ≤
≤ ≤
=
≤ ≤
≤ ≤
where
m
0=
m r
1=
6(2
p
+
3 )
q
λ
andn
0=
n r
2=
(2
p
+
3 )
q
λ
.We will construct a
P
7−
factorization ofλ
K
m n, . InP
7-factor ofλ
K
m n, , we havet
=
(
m
+
n
) 7
=
(2
p
+
3 )
q
2λ
number of vertex disjoint copies, where
a
=
2 (2
p
p
+
3 )
q
λ
, will be of type M andb
=
3 (2
q
p
+
3 )
q
λ
type W. Here type M denotesP
7-factor with its ends point in Y, and type W with its end point in X. For each1
≤ ≤
i
p
,
let{
, (2 3 )( 1) 3(2 3 ) 8( 1) 4 , 2 1:1
2
3
,1
3, 0
1, 0
1
}
i i j p q u p q v i v u w j i w
E
=
x
+ + − λ+ + λy
− + + + + − λ+ λ≤ ≤
j
p
+
q
λ
≤ ≤
u
≤ ≤
v
≤
w
≤
- . &/0, &/01
© 2011, IJMA. All Rights Reserved 2795 let
{
2( 1)
, ((2 3 ) )( 1) 3((2 3 ) ) ((2 3 ) )
p i
p
i
w t j
p q
v
p q
w
p q
t
E
+
=
x
+ − + + +
+
λ
− +
+
λ
+
+
λ
y
8p+9( 1) 3(i− + v− +1) u j, +(2p+3( 1)i− +u)λ
:1
≤ ≤
j
(2
p
+
3 )
q
λ
, 0
≤ ≤
u
2,1
≤ ≤
v
3, 0
≤
w
≤
1, 0
≤ ≤
t
1
}
.
Let
1 i p q i
,
F
E
≤ ≤ +
=
then the graphF
is aP
7-factor ofλ
K
m n, .Define a bijection fromX
∪
Y
ontoX
∪
Y
, i.e. :X
∪
Y
onto→
X
∪
Y
such that (xi, j) = xi+1, j and (yi, j) = yi+1,j where i∈
(1,2…r1) and j∈
(1,2…r2). LetF
ξ η, ={ (x) (y): x
∈
X, y∈
Y, xy∈
F}.Therefore, the graphs,
F
ξ η, {1 r1, 1 r2}, are edge disjointP
7-factor ofλ
K
m n, and its union is alsoλ
K
m n, .Thus {
F
ξ η, ; 1 r1, 1 r2} is aP
7-factorization ofλ
K
m n, .Lemma: 2.4 For any positive integer
p
andq
,
letm
=
6(3
p
+
4 )(
q p q
+
)
λ
andn
=
6(4
p
+
3 )(
q p q
+
)
λ
.
Thenλ
K
m n, has aP
7−
factorization.Proof Let
a
=
6 (
p p q
+
)
λ
,
b
=
6 (
q p q
+
)
λ
,
r
=
(3
p
+
4 )(4
q
p
+
3 )
q
andr
1=
(3
p
+
4 ),
q
r
2=
(4
p
+
3 )
q
.Let X and Y be two partite set of
λ
K
m n, and set, 1 0
, 2 0
{
:1
,1
},
{
:1
,1
}.
i j
i j
X
x
i
r
j
m
Y
y
i
r
j
n
=
≤ ≤
≤ ≤
=
≤ ≤
≤ ≤
Where
m
0=
m r
1=
6(
p
+
q
)
λ
andn
0=
n r
2=
6(
p
+
q
)
λ
.
We will construct a
P
7−
factorization ofλ
K
m n, . InP
7-factor ofλ
K
m n, , we havet
=
(
m
+
n
) 7
=
6(
p
+
q
)
2λ
number of vertex disjoint copies, where
a
=
6 (
p p q
+
)
λ
, will be of type M andb
=
6 (
q p q
+
)
λ
,
type W. Here type M denotesP
7-factor with its ends point in Y, and type W with its end point in X.For each
1
≤ ≤
i
p
,
let{
3( 1) , 4( 1) , 6( 1) 2( 1):1
6(
)
,1
3, 0
1
}
.
i i u j i u v j i u v
E
=
x
− +y
− + + + − λ+ − λ+ λ≤ ≤
j
p
+
q
λ
≤ ≤
u
≤ ≤
v
And for each
1
≤ ≤
i
q
,
let{
3 4( 1) , 4 3( 1) ,6 6( 1) 2( 1):1
6(
)
,1
3, 0
1
}
.
p i p i u v j p i u p j i u v
E
+=
x
+ − + +y
+ − + λ+ + − λ+ − λ+≤ ≤
j
p
+
q
λ
≤ ≤
u
≤ ≤
v
Let
1 i p q i
,
F
E
≤ ≤ +
=
then the graphF
is aP
7-factor ofλ
K
m n, . Define a bijection fromX
∪
Y
ontoX
∪
Y
,i.e. :
X
∪
Y
onto→
X
∪
Y
such that (xi, j) = xi+1, j and (yi, j) = yi+1,j where i∈
(1,2…r1) and j∈
(1,2…r2). Let,
F
ξ η= { (x) (y): x∈
X, y∈
Y, xy∈
F}.Therefore, the graphs
F
ξ η,{1
≤ ≤
ξ
r
1,1
≤ ≤
η
r
2},
are edge disjointP
7-factor ofλ
K
m n, and its union is alsoλ
K
m n, .Thus {
F
ξ η, ;1
≤ ≤
ξ
r
1,1
≤ ≤
η
r
2} is aP
7-factorization ofλ
K
m n, .- . &/0, &/01
© 2011, IJMA. All Rights Reserved 2796
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P
2p-factorization of a complete bipartite graph, discrete math.120 (1993), 307-308.[2] Beiling Du:
P
2k-factorization of complete bipartite multi graph. Australasian Journal of Combinatorics 21(2000), 197 - 199.[3] Ushio K: G-designs and related designs, Discrete Math., 116(1993), 299-311.
[4] Ushio K:
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3- factorization of complete bipartite graphs. Discrete math.72 (1988) 361-366.[5] Wang J and Du B:
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5- factorization of complete bipartite graphs. Discrete math. 308 (2008) 1665 – 1673.[6] Wang J:
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7-factorization of complete bipartite graphs. Australasian Journal of Combinatorics, volume 33 (2005), 129-137.[7] U. S. Rajput and Bal Govind Shukla:
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9−
factorization of complete bipartite graphs. Applied Mathematical Sciences, volume 5(2011), 921- 928.[8] Wang J and Du B:
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3 - factorization of complete bipartite multigraphs and symmetric complete bipartite multi-digraphs. Utilitas Math. 63 (2003) 213-228.
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